Left Termination of the query pattern gopher_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 MRRProof (⇔)
↳14 QDP
↳15 PisEmptyProof (⇔)
↳16 YES

(0) Obligation:

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).

Queries:

gopher(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

gopher1(nil, nil).
gopher1(cons(nil, T4), cons(nil, T4)).
gopher1(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))).
gopher1(cons(cons(cons(T34, T35), T36), T37), T39) :- gopher1(cons(T34, cons(T35, cons(T36, T37))), T39).

Queries:

gopher1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
gopher1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(cons(nil, T4), cons(nil, T4)) → gopher1_out_ga(cons(nil, T4), cons(nil, T4))
gopher1_in_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))) → gopher1_out_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23)))
gopher1_in_ga(cons(cons(cons(T34, T35), T36), T37), T39) → U1_ga(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
U1_ga(T34, T35, T36, T37, T39, gopher1_out_ga(cons(T34, cons(T35, cons(T36, T37))), T39)) → gopher1_out_ga(cons(cons(cons(T34, T35), T36), T37), T39)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2)  =  gopher1_in_ga(x1)
nil  =  nil
gopher1_out_ga(x1, x2)  =  gopher1_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(cons(nil, T4), cons(nil, T4)) → gopher1_out_ga(cons(nil, T4), cons(nil, T4))
gopher1_in_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))) → gopher1_out_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23)))
gopher1_in_ga(cons(cons(cons(T34, T35), T36), T37), T39) → U1_ga(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
U1_ga(T34, T35, T36, T37, T39, gopher1_out_ga(cons(T34, cons(T35, cons(T36, T37))), T39)) → gopher1_out_ga(cons(cons(cons(T34, T35), T36), T37), T39)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2)  =  gopher1_in_ga(x1)
nil  =  nil
gopher1_out_ga(x1, x2)  =  gopher1_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → U1_GA(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))), T39)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(cons(nil, T4), cons(nil, T4)) → gopher1_out_ga(cons(nil, T4), cons(nil, T4))
gopher1_in_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))) → gopher1_out_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23)))
gopher1_in_ga(cons(cons(cons(T34, T35), T36), T37), T39) → U1_ga(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
U1_ga(T34, T35, T36, T37, T39, gopher1_out_ga(cons(T34, cons(T35, cons(T36, T37))), T39)) → gopher1_out_ga(cons(cons(cons(T34, T35), T36), T37), T39)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2)  =  gopher1_in_ga(x1)
nil  =  nil
gopher1_out_ga(x1, x2)  =  gopher1_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
GOPHER1_IN_GA(x1, x2)  =  GOPHER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6)  =  U1_GA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → U1_GA(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))), T39)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(cons(nil, T4), cons(nil, T4)) → gopher1_out_ga(cons(nil, T4), cons(nil, T4))
gopher1_in_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))) → gopher1_out_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23)))
gopher1_in_ga(cons(cons(cons(T34, T35), T36), T37), T39) → U1_ga(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
U1_ga(T34, T35, T36, T37, T39, gopher1_out_ga(cons(T34, cons(T35, cons(T36, T37))), T39)) → gopher1_out_ga(cons(cons(cons(T34, T35), T36), T37), T39)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2)  =  gopher1_in_ga(x1)
nil  =  nil
gopher1_out_ga(x1, x2)  =  gopher1_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
GOPHER1_IN_GA(x1, x2)  =  GOPHER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6)  =  U1_GA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))), T39)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(cons(nil, T4), cons(nil, T4)) → gopher1_out_ga(cons(nil, T4), cons(nil, T4))
gopher1_in_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23))) → gopher1_out_ga(cons(cons(nil, T22), T23), cons(nil, cons(T22, T23)))
gopher1_in_ga(cons(cons(cons(T34, T35), T36), T37), T39) → U1_ga(T34, T35, T36, T37, T39, gopher1_in_ga(cons(T34, cons(T35, cons(T36, T37))), T39))
U1_ga(T34, T35, T36, T37, T39, gopher1_out_ga(cons(T34, cons(T35, cons(T36, T37))), T39)) → gopher1_out_ga(cons(cons(cons(T34, T35), T36), T37), T39)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2)  =  gopher1_in_ga(x1)
nil  =  nil
gopher1_out_ga(x1, x2)  =  gopher1_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6)  =  U1_ga(x1, x2, x3, x4, x6)
GOPHER1_IN_GA(x1, x2)  =  GOPHER1_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37), T39) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))), T39)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
GOPHER1_IN_GA(x1, x2)  =  GOPHER1_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37)) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(13) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

GOPHER1_IN_GA(cons(cons(cons(T34, T35), T36), T37)) → GOPHER1_IN_GA(cons(T34, cons(T35, cons(T36, T37))))


Used ordering: Polynomial interpretation [POLO]:

POL(GOPHER1_IN_GA(x1)) = 2·x1   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   

(14) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) YES